According to the definition of \(X_i\) , all elements are ordered in real line. Hence, \(\forall x_1, x_2, x_3 \in X_i\) , we have
\(x_1\le x_1\)  (reflexive); if \(x_1\le x_2,x_2\le x_1\) , then \(x_1=x_2\)  (antisymmetric); if \(x_1\le x_2,x_2\le x_3\) , then \(x_1\le x_2\le x_3\)  (transitive). Thus, \(X_i\)  is a partially ordered set. Besides, the join and meet of \(\forall x_1, x_2 \in X_i\)  satisfy \(x_1 \cup x_2 =\max (x_1, x_2)\in X_i\)  and \(x_1 \cap x_2 =\min (x_1,x_2)\in X_i\)  [1]}. Hence, \(X_i\)  is a lattice. We omit the proof of \(Y_k\)  for reducing repetition since they are very similar.
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